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Theorem calemos 2322
 Description: "Calemos", one of the syllogisms of Aristotelian logic. All φ is ψ (PaM), no ψ is χ (MeS), and χ exist, therefore some χ is not φ (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
calemos.maj x(φψ)
calemos.min x(ψ → ¬ χ)
calemos.e xχ
Assertion
Ref Expression
calemos x(χ ¬ φ)

Proof of Theorem calemos
StepHypRef Expression
1 calemos.e . 2 xχ
2 calemos.min . . . . . . 7 x(ψ → ¬ χ)
32spi 1753 . . . . . 6 (ψ → ¬ χ)
43con2i 112 . . . . 5 (χ → ¬ ψ)
5 calemos.maj . . . . . 6 x(φψ)
65spi 1753 . . . . 5 (φψ)
74, 6nsyl 113 . . . 4 (χ → ¬ φ)
87ancli 534 . . 3 (χ → (χ ¬ φ))
98eximi 1576 . 2 (xχx(χ ¬ φ))
101, 9ax-mp 5 1 x(χ ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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